Take the following inductive inference.

a is *y*, b is *y*, c is *y*, d is *y* |- every *x* is *y*

Now define a new predicate *z* as follows: if *x* is equal to a, b, c, or d, and *x* is *y*, then *x* is *z*, but if *x* is not equal to a, b, c, or d, and *x* is *y*, then *x* is not-*z*.

Note the equivalence.

a is *y*, b is *y*, c is *y*, d is *y* = a is *z*, b is *z*, c is *z*, d is *z*

Equivalence means that either can be substituted for the other. Now take the following inductive inference.

a is *z*, b is *z*, c is *z*, d is *z* |- every x is *z*

Substitute

a is *y*, b is *y*, c is *y*, d is *y* |- every x is *z*

Therefore, from the same premises both 'every *x* is *y*' and 'every *x* is *z*' can be induced, but both also contradict each other.

Moreover, both *y* and *z* can be defined in terms of one another. For example, take the induction.

a is *z*, b is *z*, c is *z*, d is *z* |- every *x* is *z*

Now define *y* as follows: if x is equal to a, b, c, or d, and x is z, then x is y, but if x is not equal to a, b, c, or d, and x is z, then x is not-y.

The equivalence holds as before.

a is *z*, b is *z*, c is *z*, d is *z* = a is *y*, b is *y*, c is *y*, d is *y*

And therefore, so does the substitution.

a is *y*, b is *y*, c is *y*, d is *y* |- every x is *y*

Substitute

a is *z*, b is *z*, c is *z*, d is *z* |- every x is *y*

Since the new predicate can be defined however we want, the inductive content, that is, everything entailed by everything which can be induced, must contain everything which does not contradict the premises, that is, their deductive content.

In consequence, inductive inference is not a kind of reasoning or logic, because it fails to divide the set of possible inferences over and above that which is achieved deductively.

Any thoughts?